Open Access
May 2018 Wavelet estimation for operator fractional Brownian motion
Patrice Abry, Gustavo Didier
Bernoulli 24(2): 895-928 (May 2018). DOI: 10.3150/15-BEJ790


Operator fractional Brownian motion (OFBM) is the natural vector-valued extension of the univariate fractional Brownian motion. Instead of a scalar parameter, the law of an OFBM scales according to a Hurst matrix that affects every component of the process. In this paper, we develop the wavelet analysis of OFBM, as well as a new estimator for the Hurst matrix of bivariate OFBM. For OFBM, the univariate-inspired approach of analyzing the entry-wise behavior of the wavelet spectrum as a function of the (wavelet) scales is fraught with difficulties stemming from mixtures of power laws. Instead we consider the evolution along scales of the eigenstructure of the wavelet spectrum. This is shown to yield consistent and asymptotically normal estimators of the Hurst eigenvalues, and also of the eigenvectors under assumptions. A simulation study is included to demonstrate the good performance of the estimators under finite sample sizes.


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Patrice Abry. Gustavo Didier. "Wavelet estimation for operator fractional Brownian motion." Bernoulli 24 (2) 895 - 928, May 2018.


Received: 1 January 2015; Revised: 1 September 2015; Published: May 2018
First available in Project Euclid: 21 September 2017

zbMATH: 06778351
MathSciNet: MR3706780
Digital Object Identifier: 10.3150/15-BEJ790

Keywords: operator fractional Brownian motion , operator self-similarity , Wavelets

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

Vol.24 • No. 2 • May 2018
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